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IB Mathemathics SL Review of Chapter 13: VectorsProblems Based on IB Mathematics SL Section A

Name Period

1. Given that u =C~)and v = - i , find u - v}u .- -. Given that IOAI = 3, lOBI = 5, and OA~OB = 7, (a) fmd the measure of AOB; (b) Then;._ sketch MOB and find IABIp . Given that u = (-;) and v = (;), find .C a) w such that u v W = 0; (b) the value of theconstant k such that u kv is parallel to the x-axis.

4. Two vectors oflength 5 and 6 units respectively make an angle of 50 with each other. Findthe length of the resultant.

'5. Given that a = ) and b = ~), find (i) the vector d such that d = a b ; (ii) a vector csuch that c bisects the acute angle between a and b.

6. Given that u = ( 1 21 ) and v = (-;), find the value of k such that u kv is perpendicular to v.-7. Given that u = i and v = C ~ ) ,and that point A is at (-4,-2) with AB = u andCA =v(a) find the coordinates of points Band C; (b) find the area of triangle ABC.

8. Two vectors are given by p = ~) and q = k ~ 8) ,k E R Find the value of k for whichp andq are mutually perpendicular.

9. The vectors u and v are given by u = C i and v = ~).(a) Find Iu - v i . (b) Find constants x andy such that x u y v = ( 2 2 ) .- -10. The positions of points A andB are given by OA = ~) and OB = C ; (a) Find thedistance between A and B . (b) Find the size of angle AOB to the nearest tenth of a degree.1 1 . The diagram shows a parallelogram LMNO. The points Nand M y Mhave coordinates (15,8) and (23,14) respectively. The diagram is t t scale. Find (a) ON and NM; (b) ILMI and [Ol.] , 0 x

12. The vector v is given by v = 5i 4j. The line L is perpendicular to v and contains the pointwith position vector 3; 2j. Find an equation for L in the form ax by =c.13. Vectors u and v are given by u = c j ) and v = C ~ ) . (a) Find u v; (b) Find a unit vector

perpendicular to u v .14. Given two vectors a = 2; - j and b =4; 2j, find (a) la I x Ihl; (b) b - a oa15. Given that u = i 4j and v = 3; - 7j, find the angle between the vectors u and v , giving your

answer correct to the nearest tenth of a degree.

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16. Triangle OPQ has one vertex at the origin O. Vertices P and Q are.given by the position;> ;>

vectors OP = G and OQ = C ~ Find the size of POQ, giving your answer correct to thenearest tenth of a degree.

17. Given the ~ectors u = o and v = (~) , k E R, find a value for which the angle between uand v is 60, giving your answer correct to two decimal places.

18. The diagram shows the vectors (124)and C ; Find ~o;.,. the value of a giving your answer correct to 1 a _j decimal place. -----~.---9. ABeD is a rectangle and 0is the midpoint of [AD].

-;> -;> : f < : : xpress each of the following in terms of OB and OC:-;> -;. -;.

(a) BC (b) OD (c) DC20. The vectors i ,j are unit vectors along the x-axis andy-axis respectively. The vectors

u = - 5i + 3 j and v =4 i + 7 j are given. (a) Find u + 3 v in terms oi i andj .A vector w has the same direction as u +3 v , and has a magnitude of 100.(b) Find w in terms of i and j

21. The vectors u , v are given by u = - 5i + 3 j and v = 4 i + 7 j Find scalars a , b such that a u + v = 30i + (2 - b j.

22. Find a vector equation of the line passing through (-2,-10) and (3,-6). Give your answer inthe form r= p + td where t R

23. Find the projection of v = 2i + 3j in the direction of w = 4i - 2j.-;> -;.

24. Find the projection of AB in the direction of CD for A3,2 , B -7,3 , C 9,0 , D -1,-5 .25.26.

Find the magnitude of the projection of v = 6i + 2j in the direction of w = 3i + j.Copy the diagram onto graphing paper, and mark on ~ . t : I t ~ i C : Ithe copy the following po~s: -;> l : i(a) the point D such that AD = 3 AC ; j J

;> ;> A l Bl. (b) the point P such thatAP = AB - 2 AC ; -;>

-;>

-;> -;.

27.(c) the point Q such that AQ is the projection of the vector AC in the direction of AB.: ~ : ~ ~ r : : m: : ~{~rE : jf intersection of the lines with vector equations

28. Find the scalar projection of the vector v = 4i - 3j in the direction of the vector w = 7i + 24j .29. The vector equations for the two lines L I and L2 are r = 3i 2j s2i - 6j)and

r = i 3j (2 i - 2j . Find the cosine of the angle between the lines L Iand L230. A particle moves along a line with constant velocity. The particle's initial position is

A(-7, 1,3). After two seconds, the particle is at B(5, -3, -1). (a) Find (i) velocity-;>

vector AB; (ii) the speed of the particle. (b) Write down the equation of (AB).

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Answers to illMathemathics SL Review of Chapter 13: VectorsProblems Based on illMathematics SL Section A1. 582. (a) ACE ~ 62.2'-b) IABI = 2 v S ~ 4.47A

~Bo S'1 SS-.(a) (7); (b) k = 2]4. 9.98 (3 s.f.)

5. (i) ( ) (H)o. (IT any vector of form a41.6 . - 74

7. (a) B(S,-l); QO,3); (b) ~18. 69. (a) 10 (b) = 3; y = 110. (a) S V 2 (b) 72.3- -L (a) ON = e i ; NM = (~)

(b) I LM I = 17; lO L l = 1012. 5x 4y =23

14. (a) 10 (b) 1IS. 142.816, 100.6

17. -0.43 or-8.3018. 105.8 - -b) 1OC - lOB2 219. (a) OC - OB- -1(c) 20C + 20B20. (a) 7i 24j (b) 28i 96j21. a = -30; b = 30222. r = C ; o t(~)orr =e 6 t(~)23 2, I.. 5 1- 5 J24 - 38. 19 .. 51-5

16 _ 8 V l O2S. --or --V l O S26. ~ ~ K -. . - 1 - 'l~ . . . _ .

1 I\. ~. 1 i + - 1 ;+ + - - + ~ ~ + ..l; ;f' - t - - j - - - + , ~ ~ .,

27 (:)